Al-Khayyām s Scientific Revision of Algebra

Transcription

1 Al-Khayyām s Scientific Revision of Algebra Jeffrey A. Oaks 1 Key words: Arabic algebra, al-khayyām, geometry, problem solving, concept of number, continuous magnitude. Abstract Geometers in medieval Islam found algebra to be a useful tool in solving problems, but the Greek restriction of number to positive integers, adhered to by al-khayyām (and presumably others), is at odds with the more liberal inclusion of fractions and irrational roots in traditional numerical Arabic algebra. By regarding the known and unknown numbers of traditional algebra as the homogeneous measures of continuous magnitudes (and not as magnitudes themselves), al-khayyām was able to give a rigorous foundation for the art. This revision helps us understand al-khayyām s main work on algebra in the context of geometrical and numerical problem solving, and it leads us to a reassessment of his generalized notion of number explained in his commentary on Euclid. 1 University of Indianapolis, oaks@uindy.edu. Note: The Diagram from Abý Kāmil s Algebra was generated using DRaFT ( from the reproduction of the Istanbul MS in [Abý Kāmil 1986]. The diagrams from al-khayyām s works are adapted from [Rashed & Djebbar 1981], and their lettering is taken from [Rashed & Vahabzadeh 1999]. Translations from al-khayyām s works are sometimes taken from [Rashed & Vahabzadeh, 2000], sometimes adapted from that translation, and sometimes they are mine. I indicate which in footnotes. Other translations from Arabic are mine. I thank Marco Panza for his many comments on an earlier version of this paper. Suhayl 10 (2011), pp

2 48 J.A. Oaks 1. Introduction Arabic algebra began as a practical method of arithmetical problem solving. 2 At least as far back as the early ninth century merchants, surveyors, accountants, and even the judges charged with the division of estates were said to make use of the technique. What made algebra versatile is that it could be used to solve any linear or quadratic problem, provided that the parameters be converted to numbers. For instance, one worked with numerical lengths and areas when solving mensuration problems. But algebra was not just applied to practical problems. It also turned out to be effective for solving problems in theoretical geometry that could not be worked out with straight edge and compass. This is accomplished by first setting up and simplifying a cubic equation that satisfies the conditions of the problem. The terms of the equation are then reinterpreted in the context of a new diagram in which two conic sections intersect, and the solution to the original problem is a line extending to this intersection. This technique was already well established when Umar al-khayyām (1048-ca.1131) conceived a plan to systematize it. There are only twenty-five types of simplified cubic and lower degree equations, 3 so a book listing all their solutions could serve the needs of future geometers. To meet the requirements of theoretical geometry he would also need to give algebra a rigorous foundation and provide proofs to all his solutions. Al-Khayyām first announced his plan in Risāla f taqs m rub al-dā ira (Treatise on the division of a quadrant of a circle, henceforth Quadrant). In this book he solves a geometry problem via algebra. The quartic equation he sets up simplifies to x x =20x , and this is solved by intersecting a semicircle with a hyperbola. In the middle of the solution, between the simplified equation and the construction, al-khayyām gives a summary of his views on algebra. Later he wrote his famous Risāla f lbarāh n alā masā il al-jabr wa l-muqābala (Treatise on the proofs of algebra problems, henceforth Algebra). Here he realizes his project by solving all twenty-five simplified equations of degree three and less. 2 By practical I mean that the method was taught to people in preparation for careers requiring calculation. While merchants may have been unlikely to solve quadratic equations on the job, they learned algebra as part of their education. 3 There are 25 types because coefficients were required to be positive. For example, the equations x 3 +2x =4 and x 3 +4=2x were considered to be of different kinds. See 2 below.

3 Al-Khayyām s Scientific Revision of Algebra 49 Al-Khayyām s Algebra has been well known to modern historians since the publication of Franz Woepcke s edition and translation in Since then the classification of equations and the solutions to the 14 irreducible 4 cubic cases have received ample and deserved attention. But the way al- Khayyām made practical calculation rigorous and his revision of the structure of algebraic solutions has been largely overlooked. Algebraists who wrote practical textbooks admitted fractions and irrational roots as numbers, but al-khayyām remained faithful to the Greek notion that numbers consist only of positive integers (possibly excluding 1). With such a restriction his algebraic powers cannot be unknown numbers. Instead, al-khayyām regarded them as the abstraction of quantity common to continuous magnitudes. This distinction between discrete numbers and continuous magnitudes necessitates a reworking of the third stage in the solutions of problems, that of solving the simplified equation. In traditional arithmetical algebra all unknowns are numbers, so the solutions of the six simplified equations of degrees 1 and 2 are given solely as numerical recipes. Now al-khayyām finds the need to give two separate solutions, where possible, to each equation: one if the original unknown is a number, and another if it is a magnitude. I begin this article with a review of the structure of practical Arabic algebra to put al-khayyām s work in perspective. After presenting a translation of a problem from Abý Kāmil s On the Pentagon and Decagon, solved in the traditional manner, I give al-khayyām s problem from Quadrant. This is followed by an analysis of al-khayyām s program. By observing how he manipulates his terms and from comments he makes in both works we can determine the nature of al-khayyām s powers of the unknown and his modification of the structure of algebraic solutions. A corollary of this analysis is a revision of our understanding of al-khayyām s generalized concept of number. 4 Irreducible equations are those with a constant term, and thus do not reduce to one of lower degree.

4 50 J.A. Oaks 2. Arabic algebra before al-khayyām Modern accounts of Arabic algebra, especially those which describe al- Khwārizm s early ninth century Kitāb al-jabr wa l-muqābala 5 ( Book of Algebra ), tend to focus on the solutions to the six simplified quadratic and linear equations and their geometric proofs. Before putting al-khayyām s Algebra in perspective I must put al-khwārizm s six equations in the perspective of practical algebraic problem solving. Mu ammad ibn Mýsā al-khwārizm was asked by the Caliph al-ma mýn (reigned C.E.) to write a book on algebra which encompasses the fine and important parts of its calculations that people constantly require in cases of their inheritance, their legacies, their partition, their law-suits, and their trade, and in all their dealings with one another, such as the surveying of land, the digging of canals, mensuration, and other various aspects and kinds are concerned. 6 Practical mathematics does not concern itself with philosophical notions that limit numbers to positive integers, so the concept of number for these people is any positive quantity that arises in calculation, including fractions and irrational roots. Measurements of length, area, weight, or time are always numerical. This is what al-khwārizm meant when he wrote in the beginning of his Algebra When I considered what people generally want in calculating, I found that it is always a number. 7 Al-Khwārizm s treatment of algebra, like many which followed, is divided into two parts. The first part contains an explanation of the powers of the unknown, the classification and solutions of the six simplified equations (here with proofs), and rules for operating with roots and polynomials. The second part is a collection of thirty-nine worked-out problems. The purpose of the rules in the first part is to train the student to solve the problems in the second part. 8 5 The title given by Rosen (1831) and later historians is Kitāb al-mukhta½ar f isāb al-jabr wa l-muqābala ( Brief Book on Calculation by Algebra ). Now Rashed claims that the original title was simply Kitāb al-jabr wa l-muqābala ( Book of Algebra ) [al-khwārizm 2007, 9]. 6 [al-khwārizm 2007, 95]. Translation adapted from [Gutas 1998, 113], which in turn was adapted from Rosen s translation [al-khwārizm 1831, 3]. 7 [al-khwārizm 2007, 97.1; al-khwārizm 1831, 5]. Rosen s translation. 8 The Oxford MS has 40 problems, but one of them was probably added later. After the worked-out problems al-khwārizm gives a short section on the rule of three and mensuration, followed by a long collection of worked-out inheritance problems.

5 Al-Khayyām s Scientific Revision of Algebra 51 The name given to the first degree unknown in Arabic algebra is shay ( thing ), and sometimes also jidhr ( root ). Its square is called māl (literally sum of money, property ). 9 Units were counted in dirhams (singular dirham, a silver coin), min al- adad ( in number ) or as ā ād ( units ), and frequently the appellation was dropped. Our earliest text with higher powers is QusÐā ibn Lýqā s translation of Diophantus Arithmetica, written a few decades later. The third power is ka b ( cube ), and higher powers were written as combinations of māl and ka b, such as māl māl for x 4 and māl ka b for x 5. A typical equation, from al-khwārizm s problem (4), is twenty-one things and two thirds of a thing less two māls and a sixth equals a hundred and two māls less twenty things. 10 In modern notation this is x 21 6 x2 =100 +2x 2 20x. An algebraic notation emerged in the Maghreb around the 12th c. CE, but the texts of earlier algebraists, including those by al-khwārizm, Abý Kāmil, and al-khayyām, are entirely rhetorical. The solution to a problem by medieval Arabic algebra can be broken down into three stages: Stage 1: An unknown number is named in terms of the algebraic powers (usually as a thing ). The operations or conditions specified in the enunciation are then applied to arrive at an equation. Stage 2: The equation is simplified to a standard form. The six standard equations of the first two degrees are described below. Stage 3: The solution to the simplified equation is found using the prescribed procedure. Like in medieval Europe, negative numbers and zero and were not acknowledged in Arabic mathematics. Because the solutions to simplified equations take the numbers (coefficients) of the terms as parameters, no term can be subtracted. This yields the following six types of equation: (1) ax 2 =bx, (2) ax 2 =c, (3) bx =c, (4) ax 2 +bx =c, (5) ax 2 +c =bx, and (6) bx +c =ax 2. Al-Khwārizm and later algebraists show how to solve each of 9 Because there is no good single-word English translation of māl, and because the word was used in algebra in a technical sense unrelated to its quotidian meaning, I leave it untranslated. I write its plural with the English suffix: māls. 10 [al-khwārizm 2007, 163.9].

6 52 J.A. Oaks the six types with a numerical rule. Al-Khwārizm s solution to the sample type 4 equation a half māl and five roots equal twenty-eight dirhams ( 1 2 x2 +5x =28) is: So you want to complete your māl, so that it becomes whole, which is that you double it. So double it and double everything you have which is equated with it. So it yields: a māl and ten roots equals fifty-six dirhams. So halve the roots, 11 so it yields five. So multiply it by itself, so it yields twenty-five. So add it to the fifty-six, so it yields eighty-one. So take its root, which is nine. So subtract from it half the roots, which is five. So there remains four, which is the root of the māl that you wanted, and the māl is sixteen and its half is eight. 12 Beginning with al-khwārizm and his contemporary Ibn Turk, a few of the more scientifically minded algebraists include geometric proofs of the validity of these procedures. 13 There the māl is represented by a square, and its side is a thing. The geometric magnitudes in these proofs merely represent the unknown numbers. They are not identical with them. 14 Below is a sample worked-out problem from Abý Kāmil s On the Pentagon and Decagon, which is included as part of his Book of Algebra (late 9th c.). I chose this problem because it is an application of algebra to geometry done in the traditional manner. Keep in mind that the vast majority of problems solved in Arabic algebra are arithmetic questions. 11 Here the roots is short for the number of the roots, or what we call the coefficient of the first degree term. 12 [al-khwārizm 2007, ]. 13 The following authors give geometric proofs: Al-Khwārizm (#41 [M3]), Ibn Turk (#59 [M1]), Thābit ibn Qurra (#103 [M19]), Abý Kāmil (#124 [M1]) (all four from the 9th c.), al- Karaj (al-fakhr, early 11th c., #309 [M2]), and al-samaw al (12th c., #487 [M1]). Some algebraists who either give arithmetical proofs, or no proofs at all, are: Al al-sulam (10th c., #267 [M1]), al-karaj (al-kāf, early 11th c., #309 [M1]), Ibn al-yāsam n (late 12th c., #521 [M3]), Ibn Badr (13th c.?, #587 [M1]), al-fāris (d. ca. 1320, #674 [M2]), Ibn al-bannā (d. 1321, #696, [M6, M8, M11]), al-umaw (14th c., #931 [M2]), Ibn al-hā im (1387, #783 [M13]), al-kāsh (d. 1436, #802 [M1]), SibÐ al-mārid n (late 15th c., #873 [M10]), al-qala½ād (d. 1486, #865 [M3], [M7]), Ibn Ghāz (d. 1513, #913 [M2]), and al- mil (ca. 1600, #1058 [M1]). The numbering of authors and works is taken from [Rosenfeld & Ihsano lu 2003]. 14 In some of Abý Kāmil s proofs the māl is represented by a line. Also, beginning about the 12th c. there was a shift away from geometric proof in favor of arithmetical proofs.

7 Al-Khayyām s Scientific Revision of Algebra 53 [Enunciation] So if [someone] said to you: a triangle with equal sides and angles. Each of its sides is ten, and in it is a rectangle of area ten. What is the height of the rectangle? [Stage 1] An example of this is that we make the triangle triangle ABG, and the rectangle in it rectangle HRDZ. So if we wanted to know how much line HD is, which is the height of the rectangle, we make it a thing [x ]. So line BD is the root of a third of a māl [ 1 ], and likewise line ZG is the root of a third of a māl. So 3 x2 this leaves line DZ as ten less the root of a māl and a third [ ]. So you 3 x2 multiply it by line HD, which is a thing, so it yields: ten things less the root of a māl māl and a third of a māl māl equals ten dirhams [10x =10]. x4 [Stage 2] So restore the things [by] the root of a māl māl and a third of a māl māl and add its same to the dirhams. So you get: ten dirhams and the root of a māl māl and a third of a māl māl equals ten things [ x4 =10x ]. [Stage 3] So return [the root of a māl māl and a third of a māl māl] 15 to the root of a māl māl, which is a māl, which is that you multiply it by the root of three fourths. 15 The MS has everything you have for my restored text in brackets.

8 54 J.A. Oaks And you multiply everything you have by the root of three fourths. So it yields: a māl and the root of seventy five dirhams equals the root of seventy five māls [x = 75x 2 ]. So halve the root of seventy-five māls, so it yields the root of eighteen and a half and a fourth. So multiply it by itself: it yields eighteen and a half and a fourth. So cast away from it the root of seventy-five, leaving eighteen and a half and a fourth less the root of seventy five. So you take the root. So what results, you added to it the root of eighteen and a half and a fourth. So the sum is the height of the rectangle, which is line HD. 16 In stage 3 Abý Kāmil uses the procedure for the type 5 equation to find that the height of the rectangle is the number Many of the rules given in the first part of his book are applied in this problem: operations on polynomials in stage 1, operations on square roots in stages 1 and 3, and the solution to the simplified equation in stage 3. Like many algebraists, Abý Kāmil does not explain al-jabr (restoration) and almuqābala (confrontation), the steps used when necessary in stage 2. The rules and proofs for the six equations are just one part of practical Arabic algebra. The rules are given so the practitioner or student can complete stage 3 in the solution to a problem, and proofs are provided in some books as an added measure of rigor. To many modern readers these proofs are the highlight of Arabic algebra, and so they have received the bulk of attention. This imbalance has been reinforced by al-khayyām s Algebra, where our author gives only the rules, constructions, and proofs for the simplified equations of degree three and less. Because al-khayyām wrote for mathematicians already familiar with practical algebra, he had no need to teach the rules for manipulating polynomials and roots, nor to include a collection of worked-out problems illustrating the method of algebra. 3. Al-Khayyām s worked out problem Medieval Arabic mathematicians took up the challenge of geometrical calculation that they inherited from the Greeks. While classical Greek geometers wrote their calculations in purely geometric terms, there was a trend beginning in late antiquity to posit the existence of a unit segment and 16 Problem (13) [Abý Kāmil 1986, ].

9 Al-Khayyām s Scientific Revision of Algebra 55 to express calculations with language borrowed from arithmetic. 17 For example, Archimedes related the volume of a sphere to that of a cone: Every sphere is four times a cone having a base equal to the greatest circle of the <circles> in the sphere, and, <as> height, the radius of the sphere. 18 The Baný Mýsā (9th c.), by contrast, expressed the volume of a sphere as the product of half of its diameter by a third of its surface area. 19 The surface of a sphere is curved, so this product cannot be a geometric object given in position. It is instead an abstract quantity resulting from the multiplication of two other quantities. This arithmetization facilitated the application of algebra to geometry. Al- Khayyām himself relates details of the history of these applications in both of his treatises. 20 In the late ninth century al-māhān attempted to use algebra to solve the problem of cutting a sphere by a plane so that the two parts are in a given ratio, from Archimedes On the Sphere and Cylinder II.4. He reduced the problem to an equation involving cubes, māls and numbers, 21 but he could not solve it. In the next century Abý Ja far al-khāzin made a breakthrough. He reinterpreted the cube, māl, and number as geometric magnitudes, and solved the equation as a line extending to the intersection of two conic sections in a new diagram. Later geometers were then able to solve assorted solid problems with conic sections via similar transformations through algebra. Al-Khayyām, too, found algebra to be an expedient tool for solving a geometry problem in his Quadrant. He begins by reducing the main problem, that of dividing the quadrant of a circle into two parts satisfying a certain condition, to a problem of constructing a particular right triangle. This he solves with algebra. 17 [Cuomo 2000, 180; Rashed 1996, 370, 372; Rashed & Vahabzadeh 1999, 8/2000, 8; Brentjes 2008, 452]. 18 [Netz 2004, 148], his translation. That the cone is a third of the cylinder with the same base and height was already well known. 19 [Rashed 1996, 113.8]. They had just proven in the previous proposition that the surface of a hemisphere is double the area of its great circle. 20 [Rashed & Vahabzadeh 1999, , 227.9, 255.4/2000, 111, 160, 173]. 21 [Rashed & Vahabzadeh 1999, /2000, 111].

10 56 J.A. Oaks [Enunciation] So suppose [we have] triangle ABC and angle B is right. And we draw from point B a perpendicular BD to AC. And we suppose that side AB with the perpendicular BD together are equal to AC 22 [Stage 1] And we suppose that the line AD is rational in length: let it be ten. And we suppose that BD is a thing, and we multiply it by itself: it yields a māl. And we multiply ten by itself: it yields a hundred. And we sum them: they yield a hundred and a māl, which is the square on AB, as is demonstrated in 47 of I. 23 And since the ratio of AC to AB is as the ratio of AB to AD, because of the similarity of the two triangles ABC, ABD, it follows that the product of AC by AD is equal to the square on AB. So if we divided the square on AB, which is a hundred in number and a māl, by AD, which is ten, [then] the result of the division is ten in number and a tenth of a māl, which is AC. And we supposed that AC is equal to the sum of AB, BD. So the sum of AB, BD is ten in number and a tenth of a māl. We subtracted from it BD, which is the thing, leaving ten in number and a tenth of a māl less a thing, which is AB. So we multiply it by itself. It results in: a hundred in number and three māls and a tenth of a tenth of a māl māl less twenty things, and less a fifth of a cube equals a hundred in number and a māl [100+3x x4 20x 1 5 x3 =100+x 2 ]. 22 [Rashed & Vahabzadeh 1999, 247.5/2000, 169] (my translation). 23 Euclid s Elements I.47, the Pythagorean theorem.

11 Al-Khayyām s Scientific Revision of Algebra 57 [Stage 2] So one restores and confronts and collapses, 24 leaving two māls and a tenth of a tenth of a māl māl equals twenty things and a fifth of a cube [2x x4 =20x x3 ]. So one divides everything by the thing, so it results in the four lesser types [i.e. powers] in this ratio. So the result of the division is: a tenth of a tenth of a cube and two things equals a fifth of a māl and twenty in number [ x3 +2x = 1 5 x2 +20]. So one completes a tenth of a tenth of a cube by multiplying by a hundred. Likewise we multiply all of the types by a hundred. So it results in: a cube and two hundred things equals twenty māls and two thousand in number [x x =20x ]. 25 [After a long digression in which he outlines his algebraic program, al- Khayyām returns to work out stage 3. Note the switch to geometric language at this point.] [Stage 3] We return to our problem, after reviewing these preliminaries: To find a cube, which together with two hundred of its sides, is equal to twenty squares of its side with two thousand in number. We suppose line AB is equal to the number of squares, which is twenty, and line EG is two hundred, and line EH one. Thus surface HG is two hundred. And we make a square equal to surface HG as is shown in Proposition 14 of Book II. And let the side of that square be equal to AH, where AH is perpendicular to AB, and it [AH] is the root of two hundred. And AD is the result of the division if one divides the number by the number of roots, which is ten, since the number is two thousand and the number of roots is two hundred. And if one divides the two thousand by two hundred it results in ten. And DB likewise is ten I explain the phrase restoration and confrontation (al-jabr wa l-muqābala) in [Oaks & Alkhateeb 2007]. In al-khayyām s equation this entails restoring the x by x4 the subtracted 20x x3, and confronting the 3x 2 on the left with the x 2 on the right. I have not seen the term collapse (qāåa) used in the context of simplifying an equation in any other book. It may refer to collapsing the two 100 s on either side. 25 [Rashed & Vahabzadeh 1999, /2000, 169] (my translation). This last step, multiplying each term by 100, would have traditionally been considered to be part of stage 3. But in al-khayyām s algebra it fits more naturally in stage [Rashed & Vahabzadeh 1999, /2000, 174] (my translation).

12 58 J.A. Oaks [From here al-khayyām sets up the semicircle BKD and the hyperbola KDN, and he proves that line AL (= MK) solves the equation.] There is no noticeable difference in the ways Abý Kāmil and al-khayyām carry out stages 1 and 2 in their solutions. Both assign numerical values to at least one line, and the length of another line is made a thing. The equations they set up are simplified to a standard form, with no subtracted terms and

13 Al-Khayyām s Scientific Revision of Algebra 59 only one term for each power. 27 Abý Kāmil s unknowns are numbers which can take non-integer values, so the similarities in the solutions suggest that al-khayyām s unknowns are numerical as well. By observing how al- Khayyām manipulates his algebraic terms, and from the comments he gives in both treatises, we will see that in practice he does indeed work in the same numerical system as earlier algebraists, but that he regards these numbers to be the dimensionless abstraction of quantity shared by continuous magnitudes. 4. Stages 1 & 2: Working with abstract quantity Al-Khayyām remained faithful to Aristotle regarding philosophy of mathematics. He cited Aristotle in his criticism of Euclid s and Ibn al- Haytham s admission of motion to geometry, 28 and he adhered to the Stagirite s partition of the genus of quantity into discrete and continuous. On this Aristotle wrote Of quantities some are discrete, others continuous Discrete are number and language; continuous are lines, surfaces, bodies, and also, besides these, time and place. 29 Al-Khayyām makes the same distinction in his Commentary: And when they found number to be of the same genus as magnitude, because both of them were altogether divided under the genus of quantity, they looked as well for this notion in the magnitudes. Then they found in them together with these two divisions another division. That is, the magnitudes are not made up of indivisible parts, and there is no definite end to their division, as there is with number. 30 Al-Khayyām s identification of number with positive integers puts restrictions on his numerical solutions to equations in the Algebra. For the first compound quadratic equation he writes: 27 Higher powers were common in practical algebra. In other problems Abý Kāmil set up a higher degree equation which reduces to one of lower degree. For example, in problem (51) he sets up the equation x x2 7x 3 =2x 3, which reduces to x =9x. [Abý Kāmil 1986, 96.20] 28 In his Commentary [Rashed & Vahabzadeh 1999, 311/2000, 219]. 29 Categories 4b20-24, translated in [Aristotle 1963, 12]. Number is discrete because it encompasses only positive integers, and language consists of individual syllables. 30 [Rashed & Vahabzadeh 1999, 343.5/2000, 235] (their translation).

14 60 J.A. Oaks A māl and ten roots equals thirty-nine in number [x 2 +10x =39]. So multiply half the roots by its same, and add the result to the number, and subtract from the root of the outcome half the roots. So the remainder is the root of the māl. Numerically, these two conditions are necessary: the first one of them, that the number of the roots be an even number, so that it may have a moiety; and the second, that the sum of the square of half the number of the roots and the number be a square number. For otherwise the problem would then be impossible numerically. But geometrically, not a thing at all pertaining to its problems will be impossible. 31 With this framework in mind, we can now ask how al-khayyām s number, thing, māl, cube, māl māl, etc. relate to the genus of quantity. We already know that they cannot be unknown numbers because the powers can represent the measures of continuous magnitudes in the solutions to geometry problems. In fact, al-khayyām implies that they are somehow associated with continuous magnitudes in the Algebra. In the beginning of the book, just before naming the powers of the unknown, he writes: And magnitudes are a continuous quantity, of which there are four: the line, the surface, the solid and time, as it is mentioned in a general way in the Categories and in detail in First Philosophy 32 Specifically what kinds of objects al-khayyām s powers are emerges from the way he operates on them, and from comments he makes about their nature. Al-Khayyām begins his algebraic solution in Quadrant by making line AD ten. Of course ten only measures the line, it is not identical with it. He calls BD a thing, and later line AC turns out to be ten in number and a tenth of a māl. So number, thing, and māl can all measure lines. Also in stage 1 the square on AB is calculated as a hundred and a māl, so number and māl can also measure surfaces. Later on the square on AB is also found to be a hundred in number and three māls and a tenth of a tenth of a māl māl less twenty things, and less a fifth of a cube, an aggregation of the first five powers. The ways these powers are combined shows that they 31 [Rashed & Vahabzadeh 1999, 137.5/2000, 120] (1st paragraph: my translation, 2nd paragraph: their translation). Woepcke and Rashed & Djebbar suggest that al-khayyām commits an error here. It is true that an integer solution can exist if the number of roots is odd. But al-khayyām may be speaking of the calculation of the answer, not the existence of the answer. [Woepcke 1851, 42; Rashed & Djebbar 1981, 107]. 32 [Rashed & Vahabzadeh 1999, /2000, 113] (their translation). After this quote he argues against Aristotle s place as a continuous magnitude.

15 Al-Khayyām s Scientific Revision of Algebra 61 are homogeneous, and that no term is tied to a particular dimension. Homogeneity is also evident from the fact that the powers are proportional: the ratio of the number to the roots is as the ratio of the roots to the māls, and as the ratio of the māls to the cubes, and as the ratio of the cubes to the māls māls, as far as one wishes to go. 33 We glean further insight from al-khayyām s explanation of the fourth degree term: What is here called by the algebraists a māl māl is something imagined in the continuous magnitudes, and which cannot exist by any means in sensible things. 34 By sensible things he means lines, planes, and cubes. (Later we will see also that the zero degree term number is likewise abstracted by the mind.) A māl māl (and the other higher powers) is a continuous measure, but it does not correspond in a natural way to any particular dimension. Couple this with the fact that number, thing and māl can each measure magnitudes of different dimensions, and we see that all the powers of the unknown must be imagined in the continuous magnitudes during the manipulations in stages 1 and 2. They are the abstraction of quantity, stripped of dimension, which continuous magnitudes share in this genus. By regarding the algebraic unknowns as abstract quantity, al-khayyām gives a rigorous foundation to the number system used by practical mathematicians. He writes in his Commentary: those who make calculations, I mean those who make measurements, often speak of one half of the unit, of one third thereof, and of other parts, although the unit be indivisible. But they do not mean by this a true absolute unit whereof true numbers are composed. On the contrary, they mean by this an assumed unit which, in their opinion, is divisible. Then they act in whatsoever way they please in the management of the magnitudes in accordance with that divisible unit and in accordance with the numbers composed thereof; and they often speak of the root of five, of the root of the root of ten, and of other things which they constantly do in the course of their discussions, and in their constructions and their measurements. But they only mean by this a five composed of divisible units, as we have mentioned [Rashed & Vahabzadeh 1999, 123.1/2000, 113] (my translation). 34 [Rashed & Vahabzadeh 1999, /2000, 170] (adapted from their translation). Sensible things is Rashed s and Vahabzadeh s translation of wujýh (sing. wajh). This term is translated in their French edition as individus. 35 [Rashed & Vahabzadeh 1999, /2000, 253] (their translation).

16 62 J.A. Oaks Divisible units belong to continuous magnitudes, so al-khayyām suggests in this quotation that earlier algebraists, Abý Kāmil included, had unknowingly worked with the same abstract quantities which he used in his algebraic solution. 5. Stage 3: dual solutions of quadratic equations Al-Khayyām asserted in Quadrant that algebra belongs wholly to geometry: And those who think that algebra is a means to extract unknown numbers think the unthinkable; therefore you must not pay attention to those who judge appearances and are of a different opinion: on the contrary, algebra is something geometrical which is demonstrated in Book II of the Elements, in Propositions 5 and 6 thereof. 36 But later, when he wrote his Algebra, he decided that the art could be used as a tool for solving arithmetic problems. The different natures of numbers and geometric magnitudes compelled him to give two solutions to each simplified equation: a numerical rule if the original problem is in arithmetic, and a geometric construction if it is in geometry. Al-Khayyām gives arithmetical and geometric solutions to all equations of degrees 1 and 2 and for reducible cubic equations. (A cubic equation is reducible if it has no constant term, and thus can be reduced to one of lower degree.) 37 For the fourteen irreducible cubic equations he gives only the geometric solutions because he could not solve them numerically. For these equations he writes when the subject of the problem is an absolute number, it was not feasible either for us or for any one of those concerned with the art and possibly someone else will come to know it after us 38 The equations with dual solutions follow a general pattern. The rule for the numerical solution is given first, accompanied by a geometric proof that the rule is correct. Then comes the geometric solution with its proof. (Some of these elements are missing for each equation.) 36 [Rashed & Vahabzadeh 1999, /2000, 171] (their translation). 37 Three-dimensional geometric solutions are given for the reducible cubic equations, and for their numerical solutions the reader is referred back to the solution to the corresponding quadratic equation. Marco Panza also wrote about these dual solutions in [Panza 2007, 127ff]. 38 [Rashed & Vahabzadeh 1999, 125.5/2000, 114] (their translation). Numerical solutions were first found by del Ferro and Tartaglia in the 16th c.

17 Al-Khayyām s Scientific Revision of Algebra 63 The presence of dual solutions raises the question of the degree of abstraction of al-khayyām s unknowns. In an equation like a māl and ten roots equals thirty-nine in number the value of the root can be found as a magnitude or as a number. Could al-khayyām have regarded it as being a magnitude in general, possessing a nature which rises beyond the discrete and the continuous to encompass both kinds of quantity? Such a modern interpretation is unlikely. Because the arithmetical unit is distinct from the continuous unit, al-khayyām probably viewed the abstract numbers attached to continuous quantity as capable of representing the discrete numbers of arithmetic. In this way the continuous root can represent an unknown positive integer. 6. Stage 3: Geometric solutions of irreducible cubic equations For each of the fourteen irreducible cubic equations solved in the Algebra the geometric construction flows seamlessly into its proof. The construction itself utilizes only the numbers (coefficients) of the terms of the equation, while in the proof the powers are reinterpreted as the measures of geometric magnitudes of specific dimensions. Al-Khayyām describes this transition in the interlude between stages 2 and 3 in his solution in Quadrant. He first explains that only powers up to the cube can be part of such a reinterpretation. And as to things which are used by the algebraists, and which exist in sensible things and in continuous magnitudes, they are fourfold: number, thing, māl, and cube. While number is a bit tricky, the other three powers are associated with their natural geometric counterparts: As for number, number is taken as abstracted by the mind from material things; and it does not exist in sensible things, since number is a universal intelligible thing which cannot exist except when individuated by material things. And as to the thing, its rank (manzila) in relation to continuous magnitudes is the rank of the straight line. And as to the māl, its rank is that of the rectangular equilateral quadrangle whose side is that straight line to which the expression thing is applied. And the cube is a solid which is contained by six equal rectangular equilateral square surfaces And as to the māl māl, which according to the algebraists results from the product of the māl by itself, it does not have any

18 64 J.A. Oaks meaning in the continuous magnitudes, for the square being a surface, how could it be multiplied by itself? 39 Al-Khayyām proves that the line AL satisfies the equation in Quadrant in the same way he proves his solutions in his Algebra: by comparing different volumes. Each term in the equation a cube and two hundred things equals twenty māls and two thousand in number (x x =20x ) is reinterpreted as the measure of a three dimensional rectangular solid. The cube in the equation becomes the cube on AL. The two hundred things is the product of the square on LI by AL, where the area of the square is 200. The twenty māls is the product of the square on AL by AB, where AB is twenty. Last is the two thousand, which is the square on LI by AD, or 200 by 10. Homogeneity is maintained by making the numbers (coefficients) of the first and second degree terms a plane and a line respectively, and by making the 2000 the measure of a solid. The reinterpretation of the zero degree term number ( adad) among the different equations in the Algebra depends on the degree of the equation. In the proofs of the constructions for cubic equations number measures a solid, while for quadratic equations it measures a plane. Al-Khayyām writes in the Algebra And whenever we say in this treatise a number is equal to a surface, we mean by number a rectangular surface one of whose sides is a unit while the second is a line equal in measure to the given number, and And whenever we say a number is equal to a solid, we will mean by number a rectangular parallelepipedal solid whose base is the square of the unit and whose height is equal to the supposed number. 40 So while thing, māl, and cube are always manifested as 1, 2, and 3-dimensional objects respectively in the proofs, the dimension of number changes with the degree of the equation. This is why it must be abstracted by the mind from material things, just like the māl māl. Al-Khayyām was understandably uncomfortable about including number with the other geometrical degrees. On the one hand it is necessary to reinterpret number as a geometric object in the proofs. But on the other hand it does not correspond naturally to any particular degree. His ambivalence is apparent in both treatises. We saw above in Quadrant that he 39 [Rashed & Vahabzadeh 1999, /2000, 171] (adapted from their translation). Rashed & Vahabzadeh translate manzila as position in both French and English. 40 [Rashed & Vahabzadeh 1999, , 133.7/2000, 117, 118] (their translation).

19 Al-Khayyām s Scientific Revision of Algebra 65 includes number in the list of things which exist in sensible things, but then he writes that it does not exist in sensible things. In his Algebra he writes that only three degrees fall within magnitudes : root (i.e. thing ), māl, and cube (identified with side, surface, and body respectively). But immediately after he writes what is found in the works of the algebraists with respect to these equations between the four geometrical degrees, I mean absolute numbers, sides, squares, and cubes Al-Khayyām s concept of continuous number Al-Khayyām takes up the issue of the compounding (multiplication) of ratios in the third part of his Commentary. It is here that he explains his concept of number in the context of ratios of magnitudes. He begins his proof that A:B compounded with B:C is A:C by writing: We assume the unit, and we set its ratio to the magnitude G equal to the ratio of A to B. And the magnitude G should not be regarded as being a line, or a surface, or a solid, or a time. On the contrary, it should be regarded as being abstracted by the mind from these adjunct characters and as being attached to number 42 Al-Khayyām describes G with the same kind of language he used for the algebraic terms māl māl and number (and by homogeneity all the algebraic powers). All belong to the abstract scale of measurement shared by continuous magnitudes. This level of generality is necessary in the proof because al-khayyām needs to work in a dimensionless system. At one point he multiplies the magnitude D, which is homogeneous with the unit and G, by the unit. The result must be the magnitude D itself, and not an object or quantity of a higher dimension. Some historians have inferred from the context of the proof that al- Khayyām s numbers are conceived as being the measures of ratios of magnitudes. 43 But al-khayyām defines G in the context of ratios only because this particular theorem is about ratios. As we saw in the algebraic works, his numbers derive instead from the abstraction of quantity from 41 [Rashed & Vahabzadeh 1999, ,19/2000, 114] (their translation). 42 [Rashed & Vahabzadeh 1999, /2000, 253] (adapted from their translation). 43 Two accounts of al-khayyām s numbers are [Youschkevitch 1976, 87-88; Vahabzadeh 2004].

20 66 J.A. Oaks continuous magnitudes. While his numbers can be regarded as the measures of ratios, they are not ratios essentially. 8. Summary of al-khayyām s algebra Practical algebraists calculated in a number system comprising all quantities that might arise in calculation or measurement, including fractions and irrational roots. This system was well suited for the kind of geometrical problem solving practiced by al-khayyām and others, but it needed a rigorous foundation. For this al-khayyām turned to Aristotle s genus of quantity. The species that make up this genus come in two types. For Aristotle numbers (positive integers) are an example of a discrete type, because the unit in arithmetic is indivisible. Lines, surfaces, bodies, and time are continuous types, because the units in these species are divisible. Because the algebraists worked with a divisible unit, al-khayyām identified their numbers with the measures of continuous magnitudes. These continuous numbers, as I call them, are not associated with any particular species of magnitude. A number like two and a half can be the length of a line, the area of a rectangle, the volume of a cube, or the length of a span of time. They embody the abstract quality of quantity shared by continuous magnitudes. Because the numbers are not substantiated in material things (i.e. in any species), they can only be abstracted by the mind. One advantage of such an abstract system is that the numbers are closed under multiplication. Had al-khayyām worked with geometric magnitudes themselves, there would be a problem with dimension. Multiplying two lines gives a plane, not another line, and one cannot multiply two plans or two solids together at all. But the product of two continuous numbers always yields another continuous number. This property is necessary for polynomial algebra and for al-khayyām s proof in the third part of his Commentary. Al-Khayyām works with the same unbounded range of powers as in practical algebra, and with the same names: shay ( thing ) or jidhr ( root ) for the first power, māl ( sum of money ) for the second, ka b ( cube ) for the third, followed by the higher powers māl māl, māl ka b, ka b ka b, as high as one wishes to go. Near the end of his Algebra al-khayyām even

21 Al-Khayyām s Scientific Revision of Algebra 67 works with the reciprocals of the powers. 44 Being continuous numbers, the powers are homogeneous and proportional, so no power is tied to any dimension. In the course of solving a geometry problem, for instance, a māl can be the measure of a line, a plane, or a solid. In his Quadrant al-khayyām maintained that algebra can only be used to solve geometry problems. The unknowns, after all, are the abstraction of continuous measure. But later in his Algebra he wrote that algebra can be used to solve arithmetic questions, too. 45 The continuous numbers can be taken to represent positive integers as well as the measures of geometric magnitudes, though there would probably not be much demand for the former. The structure of the solution to a problem in al-khayyām s algebra differs from that in practical algebra only in stage 3: Stage 1: An unknown number, either discrete (arithmetic) or continuous (geometry), is named in terms of the algebraic powers (usually as a thing ). The operations or conditions specified in the enunciation are then applied to arrive at an equation. Stage 2: The equation is simplified to a standard form. Al-Khayyām lists the twenty-five standard polynomial equations of the first three degrees in both treatises. Stage 3: If the enunciation asks for an unknown number, then one follows the prescribed numerical solution given in the Algebra. If the original unknown is a line, then one performs the prescribed geometric construction to produce a line whose length is the thing. 46 Al-Khayyām s Algebra is mainly a guide for performing stage 3, where the nature of the unknowns as abstract, dimensionless quantity is reinterpreted either as number or geometric magnitude. For equations of 44 [Rashed & Vahabzadeh 1999, 217/2000, 156]. Al-Karaj worked with the reciprocals of the powers in his al-fakhr (early 11th c.). 45 In addition to the three geometrical degrees, time is a kind of continuous magnitude, and al- Khayyām notes that algebra could also be used to solve problems in horology [Rashed & Vahabzadeh 1999, /2000, 113]. 46 Al-Khayyām did not give solutions for equations in which the unknown is the māl or the cube.

22 68 J.A. Oaks degree one and two he gives two solutions. If the original problem is in arithmetic, one follows the arithmetical rule. If it is in geometry, one follows the geometric construction. Al-Khayyām gives geometric proofs for both kinds of solutions. He gives only geometric solutions and proofs for irreducible cubic equations because he was not able to solve them numerically. The answer to an arithmetic problem must be a number, so al-khayyām s rule produces the answer from the numbers (coefficients) of the terms in the simplified equation, just as al-khwārizm and other algebraists had done. Al- Khayyām s geometric proofs for the numerical rules follow the tradition established by al-khwārizm and Ibn Turk, and later improved by Abý Kāmil and al-karaj. The answer to a geometry problem must be given as a geometric magnitude. The geometry problems suitable for algebra are those that ask for a magnitude whose measure satisfies a certain condition. Typically this measure is named a thing, and the resulting simplified equation is solved by constructing a line whose length is the value of the thing. Al-Khayyām proves that the constructed line satisfies the equation by reinterpreting thing, māl, and cube as the line, the square on the line, and the cube on the line respectively. While al-khayyām puts no limit on the degree of the equations in stages 1 and 2, he caps the degree of simplified equations to be solved in stage 3 at the third degree. This is due to the identification of the powers of the unknown with geometric magnitudes in the proofs of the geometric solutions, as shown in section Terminology In reading over al-khayyām s mathematical works one finds the word number used with different meanings in different contexts, and in the Algebra the names of the powers are sometimes confused with the names of their corresponding geometric magnitudes. It is necessary to distinguish between these meanings when reading his works, so I outline the potentially confusing terms here. Al-Khayyām uses the word adad ( number ) four different ways. The context of each use of the term is enough to determine which meaning he has in mind:

23 Al-Khayyām s Scientific Revision of Algebra 69 1) As the discrete objects of arithmetic, numbers are positive integers. 2) In his Commentary he speaks of his notion of number, which is abstracted by the mind as the dimensionless measure of continuous magnitudes. These continuous numbers form the foundation for al-khayyām s algebra. The powers thing, māl, etc. are regarded as unknown numbers in this sense, so the use of the word number for the zero degree term in a polynomial should be understood as taking the same meaning. 3) In the proof of the geometric solution to an equation number is reinterpreted as the measure of a solid (for cubic equations) or a plane (for quadratic equations). 4) Al-Khayyām follows earlier algebraists in speaking of the number of a term as how many there are. So the number of things in the polynomial a māl and three things is three. In modern terms, this number is the coefficient of the term. In practical Arabic algebra and in al-khayyām this number cannot be irrational, but it can be a fraction. 47 In his Quadrant al-khayyām distinguishes between the algebraic powers and their geometric counterparts by his choice of terms. During stages 1 and 2 he works with the traditional algebraic words shay ( thing ), māl, ka b ( cube ), and māl māl. He begins stage 3 by looking for a geometric Åil ( side ), murabba ( square ), and muka b ( cube ), distinguishing even between the algebraic and the geometric cube. But in the Algebra al- Khayyām is not so careful with his choice of words. In his initial list of the 25 equations he uses the proper algebraic terms, but when he restates the equations before giving their solutions, he sometimes mixes the words. For the equations with numerical solutions (degree 1 and 2) he keeps the proper algebraic terms, but for the 14 irreducible cubic equations, which are solved 47 For example, Abý Kāmil multiplies the root of five by a thing to get the root of five māls, i.e. 5 x 5x 2 [Abý Kāmil 1986, 89.12]. The root is placed over the whole term to keep the coefficient rational. See [Oaks 2009], particularly 5.2.